Recall the Cramer-Wold theorem; If $(X_n)$ and $X$ are $d$ dimensional random variables, then $X_n\to_d X$ iff $c\cdot X_n\to_d c\cdot X$ for all $c\in R^d$.
Does there exist a version of this for random elements? To be concrete, let us consider Donsker's invariance principle. https://en.wikipedia.org/wiki/Donsker%27s_theorem In the notation of the Wikipedia article, $W^{(n)}$ is a random element of $D[0,1]$ and converges to $W$=standard Brownian motion, in distribution.
Now, if we suppose our underlying random variables are $d$ dimensional and let $W$ be a $d$-dimensional standard Brownian motion, then to prove $W^{(n)}\implies W$, would it be sufficient to show $c\cdot W^{(n)}\implies c\cdot W$ for all $c\in R^d$?
Or, would one have to go via the standard route of using Cramer-Wold to prove the finite dimensional distributions converge, and then verify tightness?
PS: I know that a higher dimensional DIP holds, this is just an illustrative example.